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Chapter 5: Problem 54

You try to move a heavy trunk, pushing down and forward at an angle of \(50^{\circ}\) below the horizontal. Show that, no matter how hard you push, it's impossible to budge the trunk if the coefficient of static friction exceeds 0.84

Short Answer

Expert verified
If the coefficient of static friction exceeds 0.84, it is impossible to move the trunk by pushing at an angle of 50 degrees below the horizontal, regardless of the applied force.

Step by step solution

01

Understand the Problem

We must first recall that the force of static friction is given by \(F_{s} = \mu_{s}F_{n}\), where \(F_{s}\) is the frictional force, \(\mu_{s}\) is the coefficient of static friction, and \(F_{n}\) is the normal force or the force perpendicular to the contact surface.
02

Break down the Applied Force

The applied force can be divided into two components: a vertical component \(F_{vertical} = F_{applied} \sin(50^{\circ})\) and a horizontal component \(F_{horizontal} = F_{applied} \cos(50^{\circ})\). The vertical component of the force increases the normal force while the horizontal component tries to move the trunk.
03

Analyze the Maximum Frictional Force

For the trunk to remain stationary, the horizontal component of the force, \(F_{horizontal}\), must be less than or equal to the maximum frictional force, which can be expressed as \(F_{applied} \cos(50^{\circ}) \leq \mu_{s} (mg + F_{applied} \sin(50^{\circ}))\).
04

Apply the Limiting Condition

Rearranging the inequality from Step 3 gives \(\mu_{s} \geq \frac{F_{applied} \cos(50^{\circ})}{mg + F_{applied} \sin(50^{\circ})}\). As the applied force \(F_{applied}\) increases, the relationship approaches \(\mu_{s} \geq \tan(50^{\circ})\). From Python or calculator, plug in \(\tan(50^{\circ})\) and it's value is approximately 0.84. So, the static friction must be less than or equal to 0.84 for the trunk to move.

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Most popular questions from this chapter

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